We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several recent results involving fillability, planarity, and non-separating contact type embeddings. The cobordisms are built from symplectic handles of the form Σ × D and Σ × [−1, 1] × S1, which have symplectic cores and can be attached to contact 3-manifolds along sufficiently large neighborhoods of transverse links and preLagrangian tori. We also sketch a construction of J-holomorphic foliations in these cobordisms and formulate a conjecture regarding maps induced on Embedded Contact Homology with twisted coefficients.

We prove that an extreme Kerr initial data set is a unique absolute minimum of the total mass in a (physically relevant) class of vacuum, maximal, asymptotically flat, axisymmetric data for Einstein equations with fixed angular momentum. These data represent non-stationary, axially symmetric black holes. As a consequence, we obtain that any data in this class satisfy the inequality √J ≤ m, where m and J are the total mass and angular momentum of spacetime.

Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature,...) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds defined by curvature conditions (constant or positive or negative curvature,...). By way of contrast, geometric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two fields complement each other very well; geometric analysis offers tools for solving difficult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting ambitious goals.